Alexandr S. Mishchenko

Almost, Asymptotic and Fredholm Representations of Discrete Groups

An almost representation of a group is a map from this group into the unitary group of a Hilbert space, such that the group relations hold only approximately. We give a survey of the recent results on almost representations and on their relations to asymptotic representations. Applications to K-theory of classifying spaces are also discussed.

Nondegenerate cohomology pairing for transitive Lie algebroids, characterization

The Evens-Lu-Weinstein representation (QA, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (QAor, Dor) by tensoring by orientation flat line bundle, QAor=QA⊗or (M) and Dor=D⊗∂Aor. It is shown that the induced cohomology pairing is nondegenerate and that the representation (QAor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.

K-theory from the point of view of C ∗ -algebras and Fredholm representations

These notes represent the subject of five lectures which were delivered as a minicourse during the VI conference in Krynica, Poland, “Geometry and Topology of Manifolds”, May, 2–8, 2004.